Geometry and Topology Seminar

The stable Kauffman conjecture posits that a knot in the 3-sphere is slice if and only if it admits a slice derivative. In joint work with Alexander Zupan, we prove a related statement: A knot is handle-ribbon (also called strongly homotopy-ribbon) in a homotopy 4-ball B if and only if it admits an R-link derivative. We also show that K bounds a handle-ribbon disk D in B if and only the 3-manifold obtained by zero-surgery on K admits a singular fibration that extends over handlebodies to the complement of D, generalizing a classical theorem of Casson and Gordon stated for fibered knots. I will discuss the background (e.g. what is a knot derivative?) and the motivation (e.g. which theorem of Casson and Gordon?) behind this result, and sketch the techniques used in the proof. All of this work is joint with Alexander Zupan (University of Nebraska-Lincoln), and will appear on the arXiv in the near future.